Financial Returns and E¢ciency as seen by an Artificial Technical

Financial Returns and E¢ciency as seen by
an Arti…cial Technical Analyst¤
Spyros Skouras
European University Institute
[email protected]
First Version: December 1996
This version, March 1998
)>IJH=?J
We introduce trading rules which are selected by an arti…cially
intelligent agent who learns from experience - an Arti…cial Technical Analyst. It is shown that these rules can lead to the recognition
of subtle regularities in return processes whilst reducing data-mining
problems inherent in simple rules proposed as model evaluation devices. The relationship between the e¢ciency of …nancial markets
and the e¢cacy of technical analysis is investigated and it is shown
that the Arti…cial Technical Analyst can be used to provide a quanti…able measure of market e¢ciency. The measure is applied to the DJIA
daily index from 1962 to 1986 and implications for the behaviour of
traditional agents are derived.
The research leading to this paper was supported by a grant from I.K.Y.. Thanks are
due to Buz Brock, Dave Cass, Alex Gumbel, Blake LeBaron, Mark Salmon and particularly
Ramon Marimon for useful comments and suggestions on previous versions of this paper.
Blake LeBaron is also thanked for providing the necessary data.
¤
1 Introduction
In the last few years, increasing evidence that technical trading rules can
detect nonlinearities in …nancial time series has renewed interest in technical
analysis (see e.g. Brock, Lakonishok and LeBaron, 1992 (henceforth Brock
al.),
Neftci (1991), Levich
et al.
et
(1993) and LeBaron (1998)). Based on this
evidence, much research e¤ort has also been devoted to examining whether
trading rules can be used to evaluate and create improved time-series and
theory driven returns models (e.g. Hudson
et al.
(1996), Kho (1996), Gencay
(1996)).
The term technical analysis is used in these empirical studies to refer
to the practice of investing according to well-known technical trading rules.
However, in certain areas of …nancial theory (particularly asymmetric information models) technical analysis is de…ned to be any conditioning of
expectations on past prices (e.g. Brown and Jennings (1989), Treynor and
Ferguson (1985)).
Indeed, noisy asymmetric information models in which
rational agents condition on past prices re‡ecting (a noisy signal of) each
others information provide one explanation of why technical analysis is observed. Unfortunately though, theoretical models which lead to conditioning
consistent with the
precise form
of observed technical trading rules are cur-
rently unavailable: there is as yet no positive model of investment behaviour
which leads to actions similar to those of real Technical Analysts.
This paper models Technical Analysts as agents whos’ actions are
de facto
consistent with observed technical trading rules. Our terminology therefore
will be consistent with that of researchers examining empirical aspects of
technical analysis and as such will be more narrow than that of theoretical
models. The objective however is not the modelling of Technical Analysts
per se ;
rather, it is to use our model of a Technical Analyst to derive a more
sophisticated approach to examining the properties of trading rules.
It is
somewhat surprising that some studies have found a single arbitrarily selected rule to be “e¤ective” over long periods (e.g. Brock
et al.)
given that
real Technical Analysts use di¤erent rules in di¤erent times and in di¤erent
markets. In order to truly evaluate the e¤ectiveness of technical analysis
implemented
as
we need a model of how analysts adapt to the market environ-
ment.
We provide such a model by introducing Technical Analysts who are
arti…cially intelligent agents (see e.g.
Marimon
et al.
1990).
In Section
2 technical analysis is introduced in the simple case where agents are fully
2
informed and circumstances in which it may be a rational activity are derived.
This is a necessary step so that in Section 3 we can introduce a model of a
Technical Analyst who learns from his environment - an Arti…cial Technical
Analyst. This agent chooses amongst technical trading rules and his actions
are the outcome of an explicit decision problem which formalises the loose
notion of what it means for a rule to be “good” or optimal (examples of
applications in which a metric for comparing rules is implicit are Allen F.
and Karjalainen 1996, Neely
et al :
1996, Pictet
et al.
Allen P. and Phang 1994, Chiang 1992, Pau 1991).
1996, Taylor 1994,
This formalisation is
important because it indicates that an explicit measure of rule optimality
can and should be derived from a speci…c utility maximisation problem and
that a rule which is optimal for di¤erent types of investors (in terms of riskaversion and budget constraints) will not usually exist.
A standard application of arti…cially intelligent agents is to design them
so that their actions can reveal interesting aspects of the environment in
which they are placed (see Sargent 1993, pp. 152-160). In this vein, we will
use our Arti…cial Technical Analysts to reveal certain regularities in …nancial
data.
In particular, in Section 4 they will be used to characterise …nancial
series as in Brock
et al.,
showing that they can provide characterisations
which are more convincing and powerful than those previously attainable.
In Section 5 we propose a notion of weak market e¢ciency which is de…ned
almost exclusively as a time series property on prices and show that empirical
tests of this condition can be based on the returns obtained by the Arti…cial
Technical Analyst.
This is a step in addressing the relationship between
market e¢ciency and the pro…tability of technical analysis, an issue that has
appeared in some of the theoretical literature (e.g.
Brown and Jennings,
1989) but is absent from many recent empirical investigations of technical
analysis.
Section 6 closes this paper with a synopsis of its conclusions. The main
…nding is that the Arti…cial Technical Analyst can provide evidence corroborating the view many Technical Analysts hold of econometric returns models
and market e¢ciency: that the former are inadequate and that the latter is
not always present.
3
2 Technical analysis with full information
At a certain level of abstraction, technical analysis is the selection of rules
determining (conditional on certain events) whether a position in a …nancial
asset will be taken and whether this position should be positive or negative.
One important di¤erence between an analyst and a utility maximising investor is that the rules the analyst follows do not specify the magnitude of
the positions he should take.
This leads us to the following description of technical analysis:
Df.1: Technical Analysis is the selection of a function d which maps
the information set I at t to a space of investment decisions ­:
Ass. 1: The space of investment decisions ­ consists of three events
­ ´{Long, Neutral, Short}: It is a central feature of technical analysis that
the magnitude of these positions are undetermined and hence must be treated
as constant.
It is crucial to stress that this de…nition of technical analysis captures the
features of technical analysis as practiced. As mentioned in the introduction,
it is consistent with what is referred to as technical analysis in the literature
on empirical properties of trading rules. It is not consistent with the de…nition of technical analysis used in the literature on …nancial equilibrium with
asymmetric information where any agent who conditions on past prices is
often called a technical analyst (e.g. Brown and Jennings 1989, Treynor and
Ferguson (1985)). The reason we do not allow a more general de…nition is
because we wish to examine the properties of observed trading rules which
at …rst sight seem very di¤erent to the investment behaviour we would expect from utility maximising agents. Technical trading rules as de…ned here
have also been referred to as “market timing” rules for which an equilibrium
analysis has been developed by Merton (1981).
The event space on which these conditions are written are usually the
realisations of some random variable such as prices, volatility measures, or
the volume of trading (e.g. Blume et al. 1994) of an asset. Here, we focus
our attention on rules which are de…ned on the realisation of a history of past
prices. Restricting the information set of Technical Analysts to past prices
rather than, say, past volume is justi…ed by the fact that in order to judge
the e¤ectiveness of any rule, prices at which trade occurs must necessarily
be known. Hence, the restriction we will make allows an examination of
technical analysis when the minimum information set consistent with its
feasibility is available.
J
4
Ass. 2: I = P = fP ; P ¡1; P ¡2; :::; P ¡N +1g:
Where Pt refers to prices at t; henceforth, we use Et(¢) to refer to E(¢=It):
J
J
J
J
J
J
Technical Trading Rules and Rule Classes.
We now impose some structure on the form that the functions Pt ! ­
take, based on observation of how technical analysis is actually conducted. It
is the case that rules actually used di¤er over time and amongst analysts, but
are often very similar and seem to belong to certain “families” of closely related rules, such as the “moving average” or “range-break” family (see Brock
et al.). These families belong to even larger families, such as those of “trendfollowing” or “contrarian” rules (see for example Lakonishok et al. 1993).
Whilst it is di¢cult to argue that use of any particular rule is widespread,
certain “families” are certainly very widely used. When we choose to analyze
the observed behaviour of technical analysts, we will therefore need to utilise
the concept of a rule family, because empirical observation of a commonly
used type of function occurs at the level of the family rather than that of the
individual rule. We formalise the distinction between a rule and a family by
de…ning and distinguishing technical trading rules and technical trading rule
classes.
Df. 2: A Technical Trading Rule Class is a single valued function
D : Pt £ x ! ­ where x is a parameter vector in a parameter space B
(x 2 B µ 4k ; Pt 2 4N ).
Df. 3: A Technical Trading Rule is a single valued function:
+
dt = D(Pt ; x =x) : Pt ! ­
which determines a unique investment position for each history of prices .
2.1 Technical Analysis and rationality
Having de…ned the main concepts required to describe technical analysis,
we now attempt to identify investors who would choose to undertake this
activity. In particular, we …nd restrictions on rational (i.e. expected utility
maximising) agents’ preferences that guarantee they behave as if they were
Notice that any set of rule classes BD C =1 can be seen as a meta-class itself, where the
parameter vector N = Bi; NC determines a speci…c technical trading rule. Loosely speaking,
a rule class can be thought of as an analogue to a parametric model in econometrics and
a meta-class as a semi-parametric model.
n
i i
5
Technical Analysts. The purpose of this exercise is to clarify the meaning of
“optimal technical analysis” in a full information setting. This concept can
then be applied to the more interesting case where optimal behaviour must
be learned from experience.
For this purpose, consider the following simple but classic investment
problem. An investor i has an investment opportunity set consisting of two
assets: A risky asset paying interest R +1 (random at t) and a riskless asset
(cash) which pays no interest. He owns wealth W and his objective is to
maximise expected utility of wealth at the end of the next period by choosing
the proportion of wealth µ invested in the risky asset. We will assume µ 2
¡s; l , (s; l 2 4+ ) so that borrowing and short-selling are allowed but only to
a …nite extent determined as a multiple of current wealth. His expectations
E are formed on the basis of past prices 2 , as dictated by A :
Formally, the problem solved is:
J
J
[
]
J
2
J
max
µ2
s:t: Wt+1
=
or equivalently,
max
µ2
[¡s;l]
[¡s;l]
i
Et U (Wt+1 )
µWt(1 + Rt+1 ) + (1
¡ µ)Wt
(1)
i
Et U (Wt(1 + µRt+1 ))
the solution to which is obtained at:
µ
¤
= arg max
µ2
[¡s;l]
(2)
i
Et U (Wt(1 + µRt+1 ))
and is the solution of the …rst order condition:
0
f
g=0
Et Rt+1 U (Wt (1 + µRt+1 ))
In general therefore, µ¤ 2t ! ¡s; l is a function that depends on the
joint distribution of the random variables Rt+1 and U 0 Wt µRt+1 conditional on 2t and hence indirectly also on Wt: Rational investment behaviour
thus generally di¤ers from technical analysis in that investment behaviour
cannot be described by a function consistent with our de…nition of a trading
rule which speci…es that µ¤ may only take three distinct values and cannot
be a function of Wt.
In the special case that investors are risk-neutral however, the …rst order
condition above is inapplicable and instead µ¤ takes bang-bang solutions
:
[
]
(
6
(1 +
))
depending on sign(EJ fRJ+1 g): Denoting µ¤rn the solution to the risk-neutral
investor’s problem and assuming that µ¤rn = 0 when Et fRt+1 g = 0; then:
µ¤rn :
2t ! f ¡ s; 0; lg
which is compatible with the de…nition of a technical trading rule.
We have therefore shown that only a risk-neutral investor conditioning
on past prices will choose technical trading rules. This result is summarised
in the following proposition:
Proposition I: The risk-neutral investor solving (1) is an expected utility maximising agent who always uses technical trading
rules.
A direct corollary of this proposition is that expected utility maximisation
and technical analysis are compatible. This allows us to de…ne a technical
analyst as an expected utility maximising investor:
Df. 4. A Technical Analyst, is a risk-neutral investor who solves:
2t) ¢ Et(Rt+1)
max d(
d2 D
2J )
where D is the space of all functions with domain 4dim(
+
f¡s; 0; lg.
(3)
and image
We will let Rdt+1 ´ d(2t ) ¢ Rt+1 denote the returns accruing to an analyst
who uses a rule d: Clearly, di¤erent trading rules lead to di¤erent expected
returns.
3 Arti…cial Technical Analysts
Having speci…ed what is meant by optimal technical analysis in the case of
full information, let us assume henceforth that the technical analyst does not
know Et (Rt+1) but has a history of observations of Pt on the basis of which he
must decide his optimal action at t. This is a similar amount of information
to that possessed by econometricians and hence a technical analyst with
an e¤ective mode of learning his optimal actions in this environment is an
arti…cial intelligent agent in the sense of Sargent (1993) or Marimon et al.
(1990). The term “e¤ective” is not well de…ned in these circumstances due to
the lack of an accepted model for how learning should be conducted. Models
for learning can be justi…ed as “reasonable” in a context-dependent way (i.e.
depending on the interaction of available information and the decision in
7
which it will be used) but necessarily retain some ad hockery. In this section,
we will propose a “reasonable” model for how a Technical Analyst might try
and learn his optimal actions. The term Arti…cial Technical Analyst will refer
to a Technical Analyst operating under conditions of imperfect knowledge of
his probabilistic environment but equipped with an explicit “reasonable”
mechanism for learning optimal actions.
3.1 Parametrising Analysts’ learning
Typically, the learning technique of an arti…cial agent is similar to that of an
econometrician: the agent attempts to learn the solution to (3) by selecting
a predictor R^ +1 for E (R +1) from some parametric model speci…cation (e.g.
GARCH-M). The predictor selection is made according to some statistical
…tness criterion, such as least squares or quasi-maximum likelihood: The
arti…cial adaptive agent then chooses the optimal action d¤ which solves (3)
where E (R +1) is replaced by R^ +1:
Whilst for some applications this may be a useful approach, we are forced
to depart from this methodology somewhat due to the fact that there is
signi…cant empirical evidence that statistical …tness criteria can be misleading
when applied to decision problems such as that of the Technical Analyst. For
example, Kandel and Stambaugh (1996) show that statistical …tness criteria
are not necessarily good guides for whether a regression model is useful to
a rational (Bayesian) investor. Taylor (1994) …nds that trading based on a
channel trading rule outperforms a trading rule based on ARIMA forecasts
chosen to minimise in-sample least squares because the former is able to
predict sign changes more e¤ectively than the latter . More generally, Leitch
and Tanner (1991) show that standard measures of predictor performance
are bad guides for the ability of a predictor to discern sign changes of the
underlying variable! .
J
J
J
J
J
J
RJ+1 more
RJ+1 is irrelevant for his decision
RJ+1 are independent. Hence, a
In some circumstances, the technical analyst is interested in the sign of
than in its magnitude. In particular, the magnitude of
problem if
signRJ+1 is known or if signRJ+1 and
j
j
prediction which is formulated to take into account the purpose for which it will be used
is likely to be accurate in terms of a sign-based metric. Satchell and Timmerman (1995)
show that, without severe restrictions on the underlying series, least square metrics are
not directly related to sign-based metrics.
! A number of studies of technical trading implicitly or explicitly assume away the possibility that there exists a non-monotonic relationship between the accuracy of a prediction
in terms of a metric based on least squares and a metric based on the pro…t maximisation.
8
These empirical considerations suggest that any reasonable model of analysts’ learning must take his loss function into account. One way to achieve
this would be to create Bayesian Arti…cial Technical Analysts but this would
require speci…cation of priors on F (R +1=2 ), which might be very di¢cult.
Instead, we con…ne the Arti…cial Technical Analyst to a frequentist perspective and calculate an estimator d^¤ giving in-sample optimal solutions to (3)
from a speci…ed function space D; this is then used as an estimate for d¤.
This type of inference is commonly employed in decision theoretic applications where learning is not focused on determining the underlying stochastic
environment, but in determining an action which is an optimal decision for
an agent with very little knowledge of this environment.
The main ingredients for creating an Arti…cial Technical Analyst who
learns according to the decision theoretic approach are an in-sample analogue
to (3) and an appropriate way of restricting the functional space D. The
simplest in-sample analogue to (3) is:
J
t¡1
max §i=t¡m
N2 B
J
D(2i ; N) ¢ Ri+1
(4)
The motivation for this is that if m1 §ti¡=t1¡m D(2i; N)Ri+1 converges uniformly" to E D(2t; N) Rt+1 as m
; it is also the case that the argument maximising the former expression converges to the maximiser of the
latter. In other words, we are after a consistent estimator of the solution to
(3).
Next we choose D so as to impose some restrictions on the solution to
(4) that allow regularities of the in-sample period to be captured. Having
no theory to guide us on how to make this choice we will use empirically
observed rule classes De in the hope that the reason Technical Analysts use
them is because they are useful in solving problems similar to (3). Ideally,
we would like to have a “speci…cation test” to check whether De contains the
optimal solution to (3) but it is doubtful if a process for achieving this exists.
f
¢
g
! 1
Examples are Taylor (1989a,b,c), Allen and Taylor (1989), Curcio and Goodhart (1991)
and Arthur et al. (1996) , who reward agents in an arti…cial stockmarket according to
traditional measures of predictive accuracy. When the assumption is made explicit its
signi…cance is usually relegated to a footnote, as in Allen and Taylor (1989), p.58 fn.,
“our analysis has been conducted entirely in terms of the accuracy of chartist forecasts
and not in terms of their pro…tability or ‘economic value’ although one would expect a
close correlation between the two”. As we have argued however, the preceding statement
is unfounded and such studies must be treated with caution.
" Conditions for this are provided in Skouras (1998).
9
These considerations lead us to de…ne an Arti…cial Technical Analyst as
follows:
Df. 6: An Arti…cial Technical Analyst is an agent who solves:
§t¡1 De(2i; x)Ri+1
max
N2B i=t¡m
(5)
where De is some empirically observed D.
Turn now to an example illustrating the mechanics of this agent that will
be useful in subsequent sections.
3.1.1 Example: An Arti…cial Technical Analyst Learns the Optimal Moving Average Rule
The moving average rule class is one of the most popular rule classes used
by technical analysts and has appeared in most studies of technical analysis
published in economics journals. For these reasons, we will use it to illustrate how an Arti…cial Technical Analyst might operate by using it as a
speci…cation for De . Let us begin with a de…nition# of this class:
Df. 7: The Moving Average rule class MA(Pt; xt) is a trading
rule class s.t. :
!
¡s ifn Pt < (1 ¡ ¸) §ni=0n+1Pt¡i n
MA(Pt; x) = $" 0 if (1 ¡ ¸) §i=0n+1Pt¡i · Pt ·n(1 + ¸) §i=0n+1Pt¡i %#
l if Pt > (1 + ¸)
where Pt = [Pt;Pt¡1;:::;Pt¡N ];
x = fn;¸g;
X = fN; ¤g ;
N = f1;2;:::N g; this is the “memory” of the MA
¤ = f¸ : ¸ ¸ 0g this is the “…lter” of the MA
Now if De = MA(Pt; x), (5) becomes:
max
§t¡1 MA(Pi;n;¸)Ri+1
n;¸ i=t¡m
§i=0 Pt¡i
n+1
(6)
(7)
As de…ned, the moving average class is a slightly restricted version of what Brock et
(1991, 1992) refer to as the “variable length moving average class” (in particular, the
restriction arises from the fact that the short moving average is restricted to have length
1).
#
al.
10
Let us …x ideas with an example. Suppose an Arti…cial Technical Analyst
solving (7) wanted to invest in the Dow Jones Industrial Average and had m
daily observations of MA(2 ;n;¸)R +1 derived from N + m + 1 observations
of P . Let N = 200 and m = 250: Suppose also that s = l = 1 (i.e. position
size cannot exceed current wealth). What do the solutions to (7) look like?
Figure I plots the answer to this question$ with observations from t=1/6/1962
till 31/12/1986% (i.e. 6157 repetitions).
E
E
J
Insert Figure 1 Here
n
o6157
Figure 1 is a plot of a sequence of rule parameters (bn¤ ; b̧ )
which
=1
are optimal in a recursive sample of 250 periods. It is di¢cult to interpret
the sharp discontinuities observed in this sequence but they suggest there is
additional structure in the series that a more intelligent Arti…cial Technical
Analyst (one with a more sophisticated learning
mechanism)
might be able
o6157
n
¤
to identify. From the sequence of solutions (nb¤ ; b̧ )
optimal rules db¤ =
=1
MA(2 ; bn¤; b̧¤) can be directly derived. These& rules are optimised in the¤
period before t, and yield out of sample returns which shall be denoted R +1
and used in subsequent sections:
We now turn to applications of the Arti…cial Technical Analyst.
¤
J
J
J
4
J
J
J
J
J
J
J
@
J
Arti…cial Technical Analysts and the distribution of returns in …nancial markets.
Much of the literature on technical trading rules has asked whether popular
types of rules such as the moving average class will yield returns in excess of
what would be expected under some null hypothesis on the distribution of
$ ¤ was discretised to ¤ =
f0 0 005 0 01 0 015 0 02g This discretisation allowed us
to solve (7) by trying all dim(N) ¢ dim(¤) = 1000 points composing the solution space
in each of the 6157 recursions. More sophisticated search methods could lead to more
intelligent Arti…cial Technical Analysts but such niceties do not seem necessary when A
is as narrow as it is in this example. Furthermore it makes conditions under which (7)
converges uniformly much weaker (see Skouras (1998)).
% This data corresponds to the third subperiod used by Brock et al. and to most of the
data used by Gencay (1996).
& It may be interesting to note that these rules correspond to what Arthur (1992) terms
“temporarily ful…lled expectations” of optimal rules.
;
:
;
:
;
:
;
:
:
D
11
returns (e.g. Brock et al., Levich and Thomas (1993), Neftci (1991), etc.).
For example, under a null that returns are a random walk, trading rules can
do no better than the market mean; if this is contradicted by the data it is
evidence that the null can be rejected. A serious criticism levied against this
practice arises from the fact that it involves a testing methodology which is
not closed. The openness derives from the ad hoc choice of rules the returns
of which are examined, since these are selected according to non-rigorous
and often implicit criteria. We therefore propose restricting our choice of
rules to those chosen by Arti…cial Technical Analysts; in this way we can
overcome the problem of arbitrariness and close the methodology for testing
hypotheses on return distributions of rule classes' .
Arti…cial Technical Analysts should allow us to avoid a grave lacuna involved in the open ad hoc approach. This is that “anomalous” results may be
coincidental. In Section 4.1, we show that analysis based on a small sample
of ad hoc rules is subject to the possibility of leading to spurious conclusions
since the distribution of mean returns across rules in a class is very diverse
and hence small samples of rules are unrepresentative .
Furthermore, we show in section 4.2 that by using the more sophisticated
rules of the Arti…cial Technical Analyst we can construct more powerful tests
of the null hypotheses.
4.1 The variation of returns across rules
In this section we show that small samples of rules are insu¢cient to generate reliable conclusions about the behaviour of rule returns even within
a relatively narrow class. It would be interesting to try and derive returns
processes under which this is the case, but this has not yet been achieved.
We must therefore rely on empirical evidence to see whether rule returns are
correlated closely enough within a class to justify using a few rules as proxies
Of course, a degree of arbitrariness remains in our selection of the rule class to be
tested. However, we have already mentioned that there exists much stronger empirical
evidence on the basis of which to choose a rule class than for any speci…c rule. The arbitrariness involved in the speci…cation of learning schemes may be an additional problem,
but overall such choices are generally considered to be robust and are certainly more robust
than choices of arbitrary rules.
The likelihood of such coincidences appearing in the literature is augmented by the fact
that published research is biased in favour of reporting “anomalies” over “regularities”.
'
12
for the behaviour of the class as a whole
.
Figure 2 shows the returns accruing to each rule belonging to the moving
average class if it were applied to the data used in Section 3.1.1.
Insert Figure 2 here
Figure 2 illustrates the inadeqaucies of the
ad hoc
approach whereby
speci…c rules are used as proxies for the distribution of expected returns of a
whole class. Notice in particular two highly prominent facts evident in this
…gure.
Firstly, the highest mean return from a rule in this class is
%
times
larger than that of the worst rule. Since the means are taken from samples
with more than 6000 observations, it is unlikely that sampling uncertainty
can account for these di¤erences.
We must conclude that returns accruing
to rules within the same class vary very signi…cantly.
Secondly, the expected returns of rules display signi…cant variance even
MA!; MA"; : within small areas of the classes’ parameter space.
The best rule is the
three period moving average with no …lter
and the worst is the
four period moving average with a 2 percent …lter
.
This is
important because most researchers choose to calculate returns for a few
rules sampled evenly from the space of all rules, re‡ecting the unfounded
implicit assumption that rules are “locally” representative.
Taken together these two observations imply that
ad hoc
rules cannot be
the basis for convincing tests of speci…cations of models for returns. Rules
must be selected according to an explicit procedure which is justi…able on
theoretical grounds. Arti…cial Technical Analysts provide such a procedure.
That this is the case is suggested by Brock et al, who say that ‘Recent results in
LeBaron (1990) for foreign exchange markets suggest that the results are not sensitive to
the actual lengths of the rules used. We have replicated some of those results for the Dow
index’, p1734, fn. The “recent results” to which Brock et al. refer are a plot of a certain
statistic of 10 rules. Apart from the fact that 10 rules constitute a small sample, the
minimum statistic is almost half the size of the maximum statistic - so it is not entirely
clear that these results support the claim made. These results can be found in LeBaron
(1998).
On the other hand, the conclusions Brock et al. draw are valid (if only by coincidence)
since the rules they chose generated slightly sub-average returns.
13
4.2 Arti…cial Technical Analysts’ rules generate above
average returns
We have illustrated the fact that the use of a small sample of rules from a
class is an unreliable way of deriving conclusions regarding the behaviour of
rule returns for the whole class. We cannot know on any a priori grounds
how large this sample must be in order for the mean returns across sampled
rules to be representative of mean returns across the class. However, we now
show that “representative” mean returns are in any case likely to be less
than those obtained by an Arti…cial Technical Analyst and hence will be less
e¤ective in rejecting hypotheses on returns distributions.
Intuitively speaking, the reasonforthisis that a technical analyst who at
chooses from , should have learned to make a better-than-average choice
of . Imposing the use of a “representative” rule by an Arti…cial Technical
Analyst would be the analogue in a decision theoretic setting of estimating
a parametric model for a time series by choosing the parameters which have
average rather than minimum least squared errors. We therefore conclude
that “representative” choices of rules cannot be expected to be as good as
the rules chosen by an agent who bases his decision on past experience.
The following table serves to empirically con…rm this reasoning. Utilising
some of the information in Brock et al. (1991, Table V) it shows that the
results reported there on the basis of various …xed rules are much weaker
than those which can be drawn by using the time-varying optimal rule
derived in Section 3.1.1. That the rules chosen by Brock et al. are “average”
across the space of all rules can be veri…ed by inspection of their position in
Figure 2.
@¤
,
J
A
@
@
R ule
Long &t
Brock et al
N (buy)
6157
N(sell)
0
Buy
0.00023
Sell
-
Buy >0
-
Sell >0
-
buy-sell
-
(50,0)
3468
2636
0.4879
2782
1985
(150,0)
3581
2424
(150,0.01)
3292
2147
(200,0)
3704
2251
(200,0.01)
3469
2049
A verage
T TR ,
© O pt.
¤ ¤ ª 6157
3313
2650
-0.0004
(-1.16108)
0.00003
(-0.70959)
-0.00012
(-1.49333)
-0.00018
(-1.67583)
-0.00016
(-1.64056)
-0.00018
(-1.66579)
-0.00011
-0.00067
(-4.57949)
0.5167
(50,0.01)
0.00036
(0.90076)
0.00053
(1.64014)
0.00037
(0.94029)
0.00035
(0.80174)
0.00037
(0.92753)
0.00038
(0.96907)
0.00037
0.00095
(3.95033)
0.00041
(1.78607)
0.00049
(1.89872)
0.00049
(2.11283)
0.00052
(2.13824)
0.00053
(2.23379)
0.00056
(2.26328)
0.00048
0.00162
(7.34848)
n ; ¸J J
d n J ; ¸ J @¤J
J=1
0.5230
0.4861
0.5205
0.4777
0.5216
0.4742
0.5173
0.4780
0.5189
0.4763
0.5337
0.4675
Note that Brock et al. (1992) reproduce only a part of this table
14
T able I: T he first row of this table indicates which rule is being used: T he rules in
parentheses represent members of the moving average class; as they constitute
specifications of pairs of nJ ; ¸J: T he second and third rows indicate the number
of buy and sell signals generated: T he fourth and fifth indicate the mean return
on days on which buy and sell signals have occured: T he next two columns report
the proportion of days in which buy or sell signals were observed in which returns
were greater than zero: F inally; the last column reports the difference between
buy and sell signals: T he numbers in parentheses report results of t ¡ tests testing
whether the numbers above them are different to zero: F or the exact tests; see Brock
et al: '' :
The table indicates that all t-ratios are much higher for the optimal rule
we have developed. This means we can reject the null hypothesis that the
returns of the DJIA are normally, identically and independently distributed!
with much greater con…dence than that o¤ered by Brock et al.’s analysis. We
expect that the optimal rule will be equally powerful as a speci…cation test for
other hypothesised distributions including those considered by Brock et al.
(AR, GARCH-M, EGARCH). However, we must leave con…rmation of this
for future research.
It is on the basis of this evidence that we propose the use of Arti…cial
Technical Analysts’ rules for model speci…cation tests. This decreases the
probability of obtaining misleading results whilst at the same time delivering
more powerful conclusions.
5 Market e¢ciency and technical trading
We often read that “If markets are e¢cient, then (technical) analysis of past
price patterns to predict the future will be useless”, (Malkiel, 1992). In
this section, we attempt to analyse the relationship between the e¢ciency
of markets and the e¢cacy of technical analysis, with a view to a formal
assessment of this statement.
At present there seems to be little consensus as to what empirical properties an e¢cient market should display (see LeRoy 1989 and Fama 1991)
consequently to the lack of an accepted equilibrium model for …nancial markets. In the context of a speci…c equilibrium model, successive re…nements
on the de…nition of Fama (1970) have led to Latham’s (1989) de…nition according to which a market is E-e¢cient with respect to an information set
The table also contains information which is su¢cient to show that the Cumby-Modest
(1987) test for market timing would, if the riskless interest rate were zero, con…rm the
ability of a technical analyst learning optimal rules to conduct market timing.
!
15
IJ
i¤ revelation of
IJ
to all market participants would leave both prices and
investment decisions unchanged.
We will now propose a de…nition of e¢ciency which has the advantage
that its testability does not hinge on the assumption of a speci…c equilibrium
model. Our de…nition is a necessary condition for a market to be E-e¢cient.
It is consistent with the weak requirement implicit in almost all de…nitions
that pro…table intertemporal arbitrage should not be possible on the basis
"
of some information set
(Ross, 1987).
In its very weakest forms, this is
interpreted as meaning that once transaction costs are included, no riskaverse agent can increase his utility by attempting to “time” the market. This
statement is so weak that some authors (for example LeRoy (1989, p.1613
fn.)) consider this notion of market e¢ciency to be non-testable. However we
shall show below that under certain assumptions this test can be empirically
implemented. For simplicity, the analysis is restricted to the case where past
prices are the information set with respect to which we evaluate e¢ciency
(weak-form).
We will refer to the version of the e¢cient market hypothesis that we
have alluded to as the Lack of Intertemporal Arbitrage (LIA) Hypothesis
and discuss its implications for technical trading rules. It will become evident
that this e¢ciency notion is formulated so that it is consistent with the idea
that if technical analysis “works”, then markets must be ine¢cient. In this
sense it formalises the e¢ciency notion to which many empirical analyses of
trading rule returns allude, yet typically leave unde…ned (e.g. Hudson
et al.
1996, Taylor 1992).
LIA is a sensible equilibrium notion if it is reasonable to assume that
there exist some agents in the market who are involved in solving (1), which
we repeat here for convenience:
max
µ 2 [¡s;l ]
( (1 + µR ))
Et U i Wt
(1)
t+1
We will say that LIA is con…rmed if knowledge of past prices does not
a¤ect the optimal actions of any agent
i
solving (1).
A desired property
e¢ciency notions have failed to deliver is a way of quantifying near-e¢ciency.
We will see that LIA allows precisely such a quanti…cation.
An exception is Olsen et al. 1992, who propose a de…nition according to which ‘e¢cient
markets... are a requirement for relativistic e¤ects and thus for developing successful
forecasting and trading models’.
"
16
Df. 5.1. The Lack of Intertemporal Arbitrage (LIA) Hypothesis
holds in a market in which there exist agents who solve (1) and have
utility functions U
7 if for all U 2 7
µ¤ (2 ; W ) = µ¤ (W ) 8 2
where
µ¤ (2 ; W ) ´ arg max EU (W (1 + µR )j2 )
2¡
in a set
t
i
t
t
t
µ
µ¤ (W ) ´ arg
t
µ
t
t
s;l ]
[
max
2[¡s;l]
t
t+1
EUW (1 + µR
t
(8)
t
t+1 ))
(9)
What this de…nition implies is that if LIA holds, the true joint density
is such that knowledge that
2 =2
t
f
f
2
RJ
2
+1 j
t+1 , i.e.
For example, suppose
2
+1 ;2J
; this is not a su¢cient condition
R
nology of Clements and Hendry (1996) that R
+1 =
RJ
RJ
in no way a¤ects the investment behav-
iour of any market participant for all
for
f
independence of
t+1 and
2
t (or in the termi-
t+1 is not predictable by
2
t ).
t is only useful for predicting third and higher order
moments of the distribution. Then in a market with mean-variance agents,
actions will not be a¤ected by knowledge of
2
t although in a market pop-
ulated with other types of agents this may be the case.
Hence, according
to our de…nition, a market is e¢cient with respect to a class of agents and
the e¢ciency of a market can be viewed as a function of the size of this
o
n
´ U : µ¤ = µ¤ 8 2 .
class. Formally, the degree of e¢ciency is determined by the size of the space
7
E
t
We may interpret the size of
respect to
2
7
E
t : as the “size” of the set
as a measure of near-e¢ciency with
7
E
increases, less agents …nd
and hence the market becomes more e¢cient with respect to
2:
t
2
t useful
We must be
careful in de…ning what we mean by increases in this setting. Here we will
simply say that
7
B
is larger than
7 7
A
if
A
½7
B
in which case the proposed
measure would imply the reasonable conclusion that market A is less e¢cient
than B. Such comparisons are relevant if there exist two markets which may
be treated as seperate on
a priori
grounds or if we wish to compare the
e¢ciency of a single market during di¤erent time periods.
To decide whether a market is e¢cient with respect to a given
derive
µ¤
and
µ
¤
for that
U
H (LIA) : µ ¤ (2 ; W ) = µ¤ (W ) 8 2
t
t
t
H (Not LIA) : µ ¤ (2 ; W ) 6= µ¤ (W )
1
we may
and evaluate the following hypothesies:
0
versus
U
i
t
t
17
t
(10)
t
some
2
t
(11)
5.1 Technical Trading Rules and LIA
5.1.1 The Arti…cial Technical Analyst provides a condition on rule
returns for testing LIA
It is one of the most important features of LIA that it can be directly tested
without any assumptions on how prices are formed. To see this, notice that
if the distributions
fRJ+1
and
fRJ+1 j2t
are known, LIA can be directly evalu-
ated, in some cases even analytically. If the distributions are unknown, one
approach for empirically testing (10-11) would be to use an estimated model
for the unknown distributions
fRJ+1
and
fRJ+1 j2t .
However, such a model
may not be feasible because in practice it may be di¢cult to forecast
Rt+1
even when it is known to be predictable (for the distinction between predictability and forecastability, see e.g. Clements and Hendry, 1996). Here we
propose an alternative testing approach based on the returns obtained by an
Arti…cial Technical Analyst. This approach is based on the fact that a suf…cient condition for LIA to be violated is that technical trading is preferred
over always being long by some investors. We show this formally below:
Proposition II. Suppose an investor maximises (unconditional)
¤
expected utility by investing a fraction of wealth µ in a risky as-¤
set (see(9)), but can increase his expected utility by investing µ
¤
according to a trading rule d(2t ) (i.e. EU i (Wt(1 + µ d(2t )Rt+1 )) >
¤
EU i (Wt (1 + µ Rt+1 )) ). Then LIA does not hold# , i.e. µ¤i (2t ; Wt ) 6=
¤
µ (Wt ) for some 2t .
Proof
The assumption, may be rewritten as:
E fE [U i (Wt (1 + µ d(2t)Rt+1))j2t ]g > E fE [U i (Wt (1 + µ Rt+1))j2t ]g
¤
¤
3 4k+ s.t. Pr(2t 3) > 0 and 2t 3
¤
¤
E [U i (Wt (1 + µ d(2t)Rt+1)) 2t ] > E [U i (Wt (1 + µ Rt+1)) 2t ]
Which implies that
9
a set
µ
2
8
j
½
Now de…ne:
µ
¤¤
=
2
j
µ d(2t ) if 2t 2
µ ¤ otherwise
¤
3
¾
Note that expectations are taken with respect to BR +1 ;2 so that @2t is a random
variable unless it is made explicit that the expectation is conditional on 2t .
#
J
18
J
Then clearly, 9 µ¤¤ 6= µ¤
[ ( (1 + µ¤¤R +1))j2 ] ¸ E [U (W (1 + µ¤R +1))j2 ]82
and the inequality is strict if 2 2 3 which directly implies (11).
qed¥
s:t:E U i Wt
t
i
t
t
t
t
t
t
We conclude that a test for LIA based on trading rules, can be obtained by
replacing (10-11) with the stronger but (in some cases) empirically veri…able
conditions:
: EU (W (1 + µ¤R +1)) · EU (W (1 + µ¤R +1)) 8d
: 9 d s.t. EU (W (1 + µ¤R +1)) > EU (W (1 + µ¤R +1))
(12)
H1
(13)
Whilst rejection of H0 is not a necessary but only a su¢cient condition
for rejection of H0; we now show that it turns out that even H0 can be
empirically rejected for some important utility function classes.
H0
0
0
i
d
t
t
i
i
t
d
t
t
t
i
t
t
0
0
5.1.2 The Risk-Neutral Case
In this case, all U 2 7 are linear and the test (12-13) becomes (assuming µ¤
is positive, i.e. E (R +1) > 0):
H0 : E (R +1) · E (R +1 ) 8 d
(14)
H1 : 9 d s:t: E (R +1) > E (R +1 )
(15)
Suppose we use the rules fd(n¤; ¸¤)g6157
=1 of the Arti…cial Technical Analyst
as derived in Section 3.1.1 and the corresponding returns R +1 to test H0 :
Then referring to the table below, we conclude that the probability that H0
(LIA) is accepted is extremely low.
t
rn
d
t
0
t
rn
d
t
0
t
t
t
t
d¤
t
rn
0
Rules Mean Return St. Dev. Pr(R +1 · R +1)
4 +1 0.0002334 0.008459 4 +1 0.000801
0.008335 8.887e-5
Table II Note that the last column was calculated conditional
on the (false) assumption that 4 +1and 4 +1are normal i.i.d.
Hence we can conclude with great con…dence that there exist intertemporal arbitrage opportunities (LIA is rejected) for risk-neutral agents investing
in the market for the DJIA index.
d¤
t
t
t
¤
d
t
t
19
d
t
¤
5.1.3 The Quadratic Case
Suppose now that 7 includes quadratic funcions. Is it still the case that LIA
is violated? The reason this may not be the case is that although there exists
a rule satisfying (15) it involves greater variance than RJ+1 and hence is not
preferred agents with quadratic utility. For example, LeRoy (1989) argues
that:
...even though the existence of serial dependence in conditional expected returns implies that di¤erent formulas for trading bonds and stock will generate different expected returns, because of risk, these alternative trading rules are utilitydecreasing relative to the optimal buy-and-hold strategies.
In order to take account of this possibility when testing for LIA, we
formulate the quadratic version of (12-13) which assumes that
½
7MV = U : U (W ) = aW 2 + bW + c; a · 0; ¡b ¸ W
¾
2a
so for all U 2 7MV it is the case that EU (x) is increasing w.r.t. E (x), and
decreasing w.r.t V ar (x):
The following proposition shows the somewhat surprising result that if
there exists a rule that mean-dominates a long position, then it will also
variance dominate it and hence LeRoy’s statement is a logical impossibility in rather general circumstances. The proposition is also interesting in
its own right, it is crucial for our purposes because it establishes general
circumstances in which the quadratic case collapses to the risk-neutral case.
Proposition III: If the trading rule d1 (2t ) mean dominates trading rule d2 (2t ) and both lead to positive expected returns
E (d1Rt+1) > E (d2Rt+1) ¸ 0
and (a): The second rule is always long, i.e. is the ‘Buy and Hold’
strategy and long positions are not smaller than short positions
d2 (2t) = l all 2t
l ¸ s
or (b): Trading rules have a binary structure and position sizes are
symmetric
d1;d2 2 f¡s; lg
l = s
20
Then the returns from rule 1 have a smaller variance than the
returns from rule 2
V (d1 RJ+1 ) < V (d2RJ+1)
Proof:
Part (a)
Let d1 ´ 1 d1
Then the assumption implies E (d1R +1) > E (lR +1) ¸ 0
l
t
Notice that d
¡ ¢2
d · 1:
i
=
i
E (d1 R +1 ) > E (R +1) ¸ 0
ª
; 0; 1 and by assumption l ¸ s hence 0 > ¡
t
©
¡
s
l
t
t
s
l
¸ ¡1 so
Hence (d R +1)2 · (R +1)2
Using the fact that V (x) = E (x2) ¡ E (x)2 it follows that:
i
t
t
¡
¢
V d1 R +1 < V (R +1 )
t
t
From which it also follows directly that
V (d1 R +1 ) < V (R +1)
t
t
Part (b) ¡ ¢
Let d ´ 1 d ; i = 1; 2
Then the assumption E (d1R +1) > E (d2R +1) ¸ 0 implies
i
i
l
t
t
E (d1 R +1 ) > E (d2 R +1 ) ¸ 0
¡ ¢
Notice that d = f¡1; 1g so d 2 = 1:
Hence (d R +1)2 = (R +1)2 ; i = 1; 2
Using the fact that V (x) = E (x2) ¡ E (x)2 it follows that:
¢
¡
¢
¡
V d1 R +1 < V d2 R +1
t
i
i
t
i
t
t
t
Hence also:
qed¥
t
V (d1 R +1 ) < V (d2 R +1 )
t
t
This Proposition is useful for establishing the two following corollaries,
but also because it provides a shortcut to ranking rules by performance in
21
terms of Sharpe Ratios - an exercise attempted by many researchers and
practitioners as an alternative to ranking by mean returns (e.g. LeBaron
1998b, Sullivan, Timmerman and White 1997). In the circumstances indicated however, Sharpe Ratios are inversely related to mean returns and such
exercises are often redundant$ .
Corollary III.1: H1 is a su¢cient condition for the rejection of
LIA (i.e. (13) ) (15)) when E (R +1) ¸ 0; if U is a mean-variance
utility function and l ¸ s.
Proof
rn
i
t
This follows from Proposition IIIa. To show it, notice that ex hypothesi:
E (R +1) > E (R +1) ¸ 0
d
t
t
So by Proposition IIIa:
V (R +1) < V (R +1)
d
t
t
These two inequalities imply also that:
E (W (1 + µ¤R +1)) > E (W (1 + µ¤R +1))
V ar(W (1 + µ¤R +1)) < V ar(W (1 + µ¤R +1))
which implies that for all U 2 7
:
U (W (1 + µ¤R +1)) > U (W (1 + µ¤R +1))¥
Hence it follows that as long as l ¸ s which is very plausible, the riskt
t
d
t
d
t
t
t
t
t
MV
t
d
t
t
t
neutral case implies the mean-variance case and therefore mean-variance investors in the DJIA would …nd knowledge of past prices useful. Indeed, note
that Proposition IIIa is empirically con…rmed in Table II.
$ Special cases of this proposition have been previously established by Skouras (1997,
1996) for l = s = . This narrow circumstance seems to have been independently examined
by Acar (1998) who shows that if rules are binary, the variance of a rule is inversely related
to its mean (as in Part b). However, his proof is erroneous because as we have shown this
holds only when E RJ > (which he does not assume).
22
5.1.4 The Risk-Averse Case
In this case
7
H
is a concave class of functions and
0 is a great deal more
complicated to test. As is usually the case, an exception arises when
J+1
and dt+1 are normally distributed.
d
t+1
t+1
d
d
( t+1)
( t+1)
0
t+1
t+1
d )
t+1
R
R
Proposition IV: If R and R are normally distributed, l ¸ s
and E R
> E R ¸ , then R stochastically dominates R
(and hence all risk-averse agents will prefer R :
Proof:
In normal environments mean-variance domination and stochastic domination are equivalent (Hanoch and Levy, 1969). This together with Proposi-
¥
tion IIIa yield the desired conclusion.
If the assumptions of Proposition IV are not known to be satis…ed, we
d
t+1
over t+1. This is shown in Proposition V below:
can reformulate (12-13) in terms of a stochastic domination criterion of
R
R
Proposition V: A su¢cient condition for (13) is that E (Rt+1 ) ¸ 0;
l ¸ s and 9 d s:t M (° ) ¸ 0 8 ° and M (° ) > 0 for at least one °; where
Z°
M (° ) ´
and fRJ+1 ;
tively.
Proof
¡1
f
x ¡ fR +1 (x)]dx
[ RJ+1 ( )
@
J
fR +1 are the marginal densities of Rt+1and Rdt+1 respec@
J
M (° ) is a su¢cient condition for:
d
EU (Rt+1
) > EU (Rt+1 ) 8 concave U
¤
Notice now that when E (Rt+1 ) ¸ 0 then µ (Wt ) ¸ 0 (see (9)) and so
¤
U (Wt(1 + µ x)) is also concave in x; since:
@ U (W (1 + µ¤x)) = W µ¤U ¸ 0
t
t
@x2
@ U (W (1 + µ¤x)) = (W µ¤)2U 00 < 0
t
@x2 t
As is well known, the assumption on
Therefore it must also be that
EU (Wt(1 + µ¤Rdt+1)) > EU (Wt(1 + µ¤Rt+1))
23
¥
qed
From this proposition we notice that (12-13) can be replaced with:
ra
H0
ra
H1
6 9 d s.t.
Z
:
:
°
¡1
9 d s:t:
Z
[fR J
+1 (x) ¡ fR@J+1 (x)]dx
°
¡1
[fR J
>
+1 (x) ¡ fR@J+1 (x)]dx
0
>
8°
0
8°
(16)
(17)
and the inequality is strict for some °
Whilst a formal statistical test of (16-18) is feasible, it is incredibly cumbersome computationally (see Tolley and Pope, 1988) especially when there
are many observations on Rt+1 and Rdt+1 . This obstacle forces us to o¤er
only a casual evaluation of whether H0 can be rejected. Such an evaluation
can be conducted by inspecting a plot
of the sample version of M^ (° ) for the
d¤
optimal moving average returns Rt+1:
Insert Figure 3 Here
Observing Figure 3, we notice that for small °; M (° ) < 0, indicating that
the minimum returns from the Arti…cial Technical Analysts’ rule resulted in
smaller returns than the minimum market return. This implies that an agent
with a utility function which greatly penalizes extremely low returns would
prefer not to use the trading rule. Hence, even without taking account
of sample uncertainty we are unable to reject LIA in the risk-averse case.
Taking sample uncertainty into account using a formal statistical procedure
cannot reverse this result, since it could only make rejection of the null more
di¢cult. We conclude that there are risk-averse agents whose action cannot
be proved to depend on knowledge of past prices using this data and the type
of procedure we propose. Clearly however there may be more powerful tests
that could lead to a di¤erent result.
5.2 E¢ciency with Transaction Costs
So far we have shown that without transaction costs, there existed an arbitrage opportunity for investors in the DJIA index who had mean-variance
utility. We now consider how the inclusion of transaction costs a¤ects these
results. First of all, transaction costs will alter Arti…cial Technical Analysts’
24
objective function. To take account of this, we let s
replace (7) with:
= l = 1 as before and
(^n (q ¡1); ¸^ (q ¡1))
(18)
¡
t
1
= arg n2max
N;¸2¤f§i=t¡m M A(2i ; n; ¸)Rt+1 ¡ c jM A(2i ; n; ¸) ¡ qt¡1 jg
J
J
J
J
where c are proportional transaction costs and qt¡1 is the position
at t ¡ 1, i.e.´
³
qt¡1 2 f¡1; 0; 1g. Beginning with q0 = 0, we set q1 = M A 2t ; n
^ t (q0) ; ¸^ t (q0)
and iterate so as to obtain the sequence fqt g : From this, we next obtain
³
¤´
n¤t ; ¸t
^ ^
¤
= (^nt(qt¡1); ¸^ t(qt¡1)) with which we derive
n
o6157
(2t; n^¤t ; ¸^¤t )
MA
t=1
and Rdt+1 for various levels of c which are proportional transaction costs: Our
objective will be to determine the level of transaction costs% c for which LIA
holds, i.e. H0rn cannot be rejected in favour of H1rn at the 95% con…dence
level& . This exercise is conceptually similar to that of Cooper and Kaplanis
(1994) who try to estimate the level of deadweight costs that would explain
the home bias in international equity portfolios. Allowing for transaction
costs changes our near-equilibrium notion in that we now seek pairs f E ; cg
rather than just E for which LIA holds.
In Table III below we have, in the second column, tabulated the returns
from a rule used by the analyst who solves (18). The level of costs at which
H0 can be rejected under the assumption that Rt+1 ; Rdt+1 » N i:i:d: is represented by the line dividing Table III. Notice that this table incorporates the
0
0
7
7
¤
Note that as de…ned, the cost of switching from a long to a short position and vice
versa is c:
& It is important to note that Proposition III can be extended to the case of transaction
costs if these are small enough. The same is not true for Proposition I if transaction costs
are proportional.
%
25
special case c
= 0 reported in Table II.
@
Pr(R@J · RJ) ¦6157
J=1 (1 + RJ )
Rules
Market RJ
Opt. TTR
c=0
Mean Return
0.0002334
St. Dev.
0.008459
-
2.378
0.000801
0.008335
8.887e-05
110.7
c=0.0001
c=0.0002
c=0.0003
c=0.0004
c=0.0005
c=0.0006
c=0.0007
c=0.0008
c=0.0009
c=0.001
0.0007304
0.0006911
0.008328
0.008321
0.0005109
0.001238
71.41
55.88
0.0006196
0.008318
0.00533
35.63
0.0005508
0.0004893
0.008311
0.008305
0.01788
0.0452
22.99
15.44
0.0004707
0.0003741
0.00829
0.00828
0.058
0.1755
13.67
7.101
0.0003099
0.008273
0.3061
4.456
0.0002763
0.0002191
0.008205
0.008215
0.3876
0.5379
3.453
2.13
Table III: The …rst column indicates which level of costs is
under consideration. The next two columns indicate the empirical
mean and the standard deviation of the rules’ returns.
The fourth column shows the probability (under the assumption
of normal distributions) that the mean returns from a speci…c rule
were smaller or equal to the mean market returns. The …nal
column shows the cumulative returns from each strategy during
the whole time period.
The table indicates that at the 5% level of signi…cance, LIA will be ac-
¸ 0 06%
· 0 09%
cepted for c
:
: The mean return of the optimal rule remains larger
for c
:
(but not for the usual con…dence margin)' . These levels of
c make it tempting to argue that with todays’ cost conditions LIA is violated. However, costs were certainly larger at the beginning of the sample we
have considered. How large the decrease in transaction costs has been and
how it has a¤ected di¤erent types of investors is a question which is beyond
the scope of this paper and which we do not attempt to answer. We must
Note that Proposition IIIa can be extended to the case with transaction costs if these
are “small enough”. Table III is consistent with the results of Proposition IIIa.
An investor with access to a discount broker, e.g. via email, can purchase 1000 shares
of a company listed on the NYSE for a $14.95 fee. However, micro-structure frictions such
as bid-ask spreads should also be taken into account.
'
26
add the warning that the time-series used is not adjusted for dividends and
hence our results are likely to be biased against LIA.
In this section, we have derived conditions on B - ;cC that ensure that
the trading rule returns of an Arti…cial Technical Analyst are such that we
cannot reject our version of the weak form of the e¢cient market hypothesis. The restrictions derived are not trivial and hence even though our
e¢ciency notion is only a special case of other standard notions and even
though we have only tested su¢cient conditions for its’ rejection, we have
found a class of agents de…ned by their preferences and transaction costs who
would optimally condition actions on past prices. Note that there are numerous asymmetric information models which generate equilibria for which
LIA is violated (e.g. Hussman 1992, Brown and Jennings 1989, Treynor and
Ferguson 1985) so some of the available models may describe the data generating process accurately. The methodology proposed is useful because it
unties e¢ciency from a speci…c equilibrium notion and allows near-e¢ciency
comparisons across time and markets by comparison of the generality of the
conditions on B - ;cC that would ensure LIA is not rejected. In thus provides
a quanti…able measure of near e¢ciency. Equally importantly, it formalises
a sense in which markets can be characterised as ine¢cient when empirical
studies …nd trading rules to be pro…table. It therefore captures an informal
notion of e¢ciency used by many researchers.
7
7
6 Conclusions
Empirical investigations of …nancial series have previously found technical
trading rules to be useful research tools. In this paper we develop a more
sophisticated form of technical analysis and show empirically that it can be
used to obtain powerful characterisations of …nancial series.
Our …rst step is to show that technical analysis is consistent with utility
maximisation since in a full information environment risk-neutral investors
will use technical trading rules. This formalisation is revealing because it
clearly illustrates that the optimality of a trading rule can only be judged in
the context of a speci…c decision problem and hence a speci…c class of rules,
a level of transaction costs, a position in the market and most importantly
a utility function. A generally optimal technical trading rule is therefore a
chimera and hence choices from any class of rules should be derived from
explicit criteria based on some decision problem.
27
Relaxing investors’ knowledge of the stochastic behaviour of returns to a
realistic level, we develop a simple model for how Technical Analysts might
behave in environments where they can learn from experience. An agent
using a simple learning model to make choices among technical trading rules
is called an Arti…cial Technical Analyst. This construct is useful because it
provides a way to empirically evaluate claims of real-world technical analysts
that …nancial time-series models are inadequate for their decisions and that
their pro…ts prove markets are not always e¢cient.
In this vein, we use the rules which are optimal for the Arti…cial Technical Analyst in empirical applications and …nd that in addition to being
subject to fewer data-mining problems than arbitrary …xed rules used in previous studies, they also lead to more powerful inferences. Consequently, we
suggest that bootstrap based model speci…cation tests based on rule returns
as pioneered by Brock et al. (1992) should be augmented with arti…cially
intelligent agents in the spirit of Sargent (1993).
Finally, we investigate the relationship between trading rule returns and
market e¢ciency. Our analysis has been based on developing and applying a
way of empirically rejecting LIA, the hypothesis that past prices do not a¤ect
investment decisions. Whilst this may seem intuitively plausible, it is a new
de…nition of e¢ciency that has the advantage of being testable independently
of an equilibrium model. It relates to previous notions in that it captures
the main intuition underlying empirical tests of weak-form e¢ciency and is a
necessary condition for some standard theoretical notions of e¢ciency such
as Latham’s (1989) E-e¢ciency. It also has the advantage that it provides a
quanti…able notion of near-e¢ciency.
Our empirical test uses the DJIA index as a proxy for the market portfolio.
For the data considered (daily 1962-1986), LIA can be rejected for agents with
mean-variance utility facing low enough transaction costs; however, there do
exist other risk-averse agents who cannot be shown to …nd conditioning on
past prices to be useful. We interpret the magnitude of transaction costs
and the generality of the preferences for which this hypothesis is rejected as
a measure of market e¢ciency. An interesting experiment which we must
leave for future investigations is the comparison of the size of this measure
to ones obtained from other …nancial series. We note that a by-product of
our analysis is the observation that under general assumptions the variance
of trading rule returns are inversely related to their mean.
There are many natural extensions of this work so we can only make
a few indicative suggestions. Firstly, the Arti…cial Technical Analyst could
28
be made more intelligent by making his learning more sophisticated and by
widening the space of trading rules from which he may choose. Secondly,
it would be important to try and …nd models for returns that are consistent with returns obtained by the Arti…cial Technical Analyst. Finally, it is
quite easy to extend the framework so as to allow for analysts who choose
rules conditional on variables other than past prices. This indicates that the
Arti…cial Technical Analyst is capable even of fundamental analysis.
29
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!"
Evolution of optimal moving average memory (n) overtime
200
150
100
50
0
0
1000
2000
3000
4000
5000
6000
7000
Evolution of optimal moving average window (lamda) overtime
0.015
0.01
0.005
0
0
1000
2000
3000
4000
5000
6000
7000
of each optimal parameter n¤ and ¸¤ respectively, during
Figure
1: Evolution
¤
£
t
J
T 5; T . :
35
J
−3
x 10
1.2
1
0.8
0.6
0.4
0.2
0
0.02
200
0.015
150
0.01
100
0.005
50
0
0
Figure 2: Mean Returns of each rule (n; ¸). The mean is taken over t
[TI ; TB ]:
36
−4
6
x 10
5
4
3
2
1
0
−1
0.94
0.96
0.98
1
1.02
1.04
?° the sample version of M ° Figure 3: This is M
37
1.06